I think a prototypical amoral economist would "solve" the problem by having people pay the cops some kind of percentage fee if they get their car back. Or at least I think that would work, but it bothers me.
Or just change the incentive structure. Make the fines go to the general government operating fund, not directly to the police department. Assess performance based on solving the crimes you want solved, rather than pure # of offenses. It's not hard to come up with alternatives.
